Towards Optimal Degree Distributions for Left-Perfect Matchings in Random Bipartite Graphs

AbstractIn this paper, we consider the relationships between the numbers of perfect matchings (1-factors) of bipartite graphs and Pell, Mersenne and Perrin Numbers. Then we give some Maple procedures in order to calculate the numbers of perfect matchings of these bipartite graphs.

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Rainbow Perfect Matchings in Complete Bipartite Graphs: Existence and Counting

Combinatorics Probability Computing ◽

10.1017/s096354831300028x ◽

2013 ◽

Vol 22 (5) ◽

pp. 783-799 ◽

Cited By ~ 4

Author(s):

GUILLEM PERARNAU ◽

ORIOL SERRA

Keyword(s):

Bipartite Graph ◽

High Probability ◽

Perfect Matching ◽

Complete Bipartite Graph ◽

Bipartite Graphs ◽

Perfect Matchings ◽

Asymptotic Enumeration ◽

Complete Bipartite Graphs ◽

Edge Colourings ◽

Square Matrix

A perfect matchingMin an edge-coloured complete bipartite graphKn,nis rainbow if no pair of edges inMhave the same colour. We obtain asymptotic enumeration results for the number of rainbow perfect matchings in terms of the maximum number of occurrences of each colour. We also consider two natural models of random edge-colourings ofKn,nand show that if the number of colours is at leastn, then there is with high probability a rainbow perfect matching. This in particular shows that almost every square matrix of ordernin which every entry appearsntimes has a Latin transversal.

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A Graph Polynomial for Independent Sets of Bipartite Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548312000296 ◽

2012 ◽

Vol 21 (5) ◽

pp. 695-714 ◽

Cited By ~ 5

Author(s):

Q. GE ◽

D. ŠTEFANKOVIČ

Keyword(s):

Riemann Hypothesis ◽

Independent Sets ◽

Bipartite Graphs ◽

Rational Points ◽

Perfect Matchings ◽

Generalized Riemann Hypothesis ◽

Graph Polynomial ◽

Exact Evaluation

We introduce a new graph polynomial that encodes interesting properties of graphs, for example, the number of matchings, the number of perfect matchings, and, for bipartite graphs, the number of independent sets (#BIS).We analyse the complexity of exact evaluation of the polynomial at rational points and show a dichotomy result: for most points exact evaluation is #P-hard (assuming the generalized Riemann hypothesis) and for the rest of the points exact evaluation is trivial.

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